Generic objects in topology Wiesaw Kubi s Institute of - - PowerPoint PPT Presentation

Generic objects in topology

Wiesław Kubi´ s

Institute of Mathematics, Czech Academy of Sciences —— College of Science, Cardinal Stefan Wyszy´ nski University in Warsaw http://www.math.cas.cz/kubis/

12th TOPOSYM Prague, 25-29.07.2016

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 1 / 24

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 2 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game.

BM (K)

Fix a subcategory K of CM+, whose arrows are surjections.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game.

BM (K)

Fix a subcategory K of CM+, whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game.

BM (K)

Fix a subcategory K of CM+, whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K. Eve starts by choosing a K-object K0.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game.

BM (K)

Fix a subcategory K of CM+, whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K. Eve starts by choosing a K-object K0. Odd responds by choosing a K-arrow k1

0 : K1 → K0.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game.

BM (K)

Fix a subcategory K of CM+, whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K. Eve starts by choosing a K-object K0. Odd responds by choosing a K-arrow k1

0 : K1 → K0.

Eve responds by choosing a K-arrow k2

1 : K2 → K1.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game.

BM (K)

Fix a subcategory K of CM+, whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K. Eve starts by choosing a K-object K0. Odd responds by choosing a K-arrow k1

0 : K1 → K0.

Eve responds by choosing a K-arrow k2

1 : K2 → K1.

And so on...

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM+ the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game.

BM (K)

Fix a subcategory K of CM+, whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K. Eve starts by choosing a K-object K0. Odd responds by choosing a K-arrow k1

0 : K1 → K0.

Eve responds by choosing a K-arrow k2

1 : K2 → K1.

And so on... The result is an inverse sequence

• k = Ki, kj

i , ω.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Who wins?

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Who wins?

BM (K, U)

Fix a compact space U.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Who wins?

BM (K, U)

Fix a compact space U. We say that Odd wins if the limit of the inverse sequence K0 K1 K2 · · ·

k1 k2

1

k3

2

is homeomorphic to U.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Who wins?

BM (K, U)

Fix a compact space U. We say that Odd wins if the limit of the inverse sequence K0 K1 K2 · · ·

k1 k2

1

k3

2

is homeomorphic to U.

Definition

We say that U is K-generic if Odd has a winning strategy in BM (K, U).

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Who wins?

BM (K, U)

Fix a compact space U. We say that Odd wins if the limit of the inverse sequence K0 K1 K2 · · ·

k1 k2

1

k3

2

is homeomorphic to U.

Definition

We say that U is K-generic if Odd has a winning strategy in BM (K, U). The game above will be called the Banach-Mazur game with parameters K and U.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Uniqueness

Proposition

A K-generic compact space (if exists) is unique, up to homeomorphisms.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness

Proposition

A K-generic compact space (if exists) is unique, up to homeomorphisms.

Proof.

1

Suppose U0, U1 are K-generic, witnessed by strategies Σ0, Σ1, respectively.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness

Proposition

A K-generic compact space (if exists) is unique, up to homeomorphisms.

Proof.

1

Suppose U0, U1 are K-generic, witnessed by strategies Σ0, Σ1, respectively.

2

Assume Odd uses Σ0.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness

Proposition

A K-generic compact space (if exists) is unique, up to homeomorphisms.

Proof.

1

Suppose U0, U1 are K-generic, witnessed by strategies Σ0, Σ1, respectively.

2

Assume Odd uses Σ0.

3

Assume Eve uses Σ1.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness

Proposition

A K-generic compact space (if exists) is unique, up to homeomorphisms.

Proof.

1

Suppose U0, U1 are K-generic, witnessed by strategies Σ0, Σ1, respectively.

2

Assume Odd uses Σ0.

3

Assume Eve uses Σ1.

4

They both win!

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness

Proposition

A K-generic compact space (if exists) is unique, up to homeomorphisms.

Proof.

1

Suppose U0, U1 are K-generic, witnessed by strategies Σ0, Σ1, respectively.

2

Assume Odd uses Σ0.

3

Assume Eve uses Σ1.

4

They both win!

5

Thus U0 ≈ U1.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Universality

Proposition

Let U be a K-generic space. Then for every K ∈ K there exists a continuous surjection q : U → K.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 6 / 24

Universality

Proposition

Let U be a K-generic space. Then for every K ∈ K there exists a continuous surjection q : U → K. If all K-arrows are retractions, then q is a retraction.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 6 / 24

Basic examples

Example

The Cantor set 2ω is CM+-generic.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 7 / 24

Basic examples

Example

The Cantor set 2ω is CM+-generic.

Proof.

Odd has a simple winning tactic: At each step he chooses an arbitrary continuous surjection from 2ω.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 7 / 24

Basic examples

Example

The Cantor set 2ω is CM+-generic.

Proof.

Odd has a simple winning tactic: At each step he chooses an arbitrary continuous surjection from 2ω.

Example

Let Fin+ be the category of nonempty finite sets with surjections. Then 2ω is Fin+-generic.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 7 / 24

Continua

Proposition

There is no generic continuum. More precisely, there is no C-generic space, where C is the category of continua with continuous surjections. a continuum = a nonempty compact connected metrizable space

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 8 / 24

Continua

Proposition

There is no generic continuum. More precisely, there is no C-generic space, where C is the category of continua with continuous surjections. a continuum = a nonempty compact connected metrizable space

Proof.

Use Waraszkiewicz spirals.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 8 / 24

The pseudo-arc

Notation:

Denote by I the category whose unique object is the unit interval I = [0, 1] and arrows are continuous surjections.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 9 / 24

The pseudo-arc

Notation:

Denote by I the category whose unique object is the unit interval I = [0, 1] and arrows are continuous surjections.

Theorem

There exists an I-generic continuum, namely, the pseudo-arc P.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 9 / 24

Domination

Definition

Let K0 be a subcategory of a category K ⊆ CM+. We say that K0 dominates K if

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 10 / 24

Domination

Definition

Let K0 be a subcategory of a category K ⊆ CM+. We say that K0 dominates K if

1

For every X ∈ Obj(K) there are X0 ∈ Obj(K0) and a K-arrow f : X0 → X.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 10 / 24

Domination

Definition

Let K0 be a subcategory of a category K ⊆ CM+. We say that K0 dominates K if

1

For every X ∈ Obj(K) there are X0 ∈ Obj(K0) and a K-arrow f : X0 → X.

2

For every ε > 0, for every K-arrow p: X → X0 with X0 ∈ Obj(K0) there exists a K-arrow q : Y0 → X with Y0 ∈ Obj(K0) such that p ◦ q is ε-close to some K0-arrow g : Y0 → X0.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 10 / 24

Domination

Definition

Let K0 be a subcategory of a category K ⊆ CM+. We say that K0 dominates K if

1

For every X ∈ Obj(K) there are X0 ∈ Obj(K0) and a K-arrow f : X0 → X.

2

For every ε > 0, for every K-arrow p: X → X0 with X0 ∈ Obj(K0) there exists a K-arrow q : Y0 → X with Y0 ∈ Obj(K0) such that p ◦ q is ε-close to some K0-arrow g : Y0 → X0. That is: (∀ y ∈ Y0) ̺(p(q(y)), g(y)) ε, where ̺ is a fixed metric on X0.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 10 / 24

Theorem

Assume K0 is dominating in K. Then K0-generic = ⇒ K-generic.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 11 / 24

Theorem

Assume K0 is dominating in K. Then K0-generic = ⇒ K-generic.

Proof.

K0 K ′ K1 K2 K ′

2

K3 · · · K ′ K1 K ′

2

K3 · · ·

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 11 / 24

Theorem

Assume K0 is dominating in K. Then K0-generic = ⇒ K-generic.

Proof.

K0 K ′ K1 K2 K ′

2

K3 · · · K ′ K1 K ′

2

K3 · · · Use Mioduszewski’s result (1963) on homeomorphisms of inverse limits.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 11 / 24

Corollary

The pseudo-arc P is generic in the class of all Peano continua.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 12 / 24

Fix a category C ⊆ CM+ whose arrows are surjections.

Definition

We say that C is directed if for every X, Y ∈ Obj(C) there exist W ∈ Obj(C) and C-arrows f : W → X, g : W → Y.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 13 / 24

Fix a category C ⊆ CM+ whose arrows are surjections.

Definition

We say that C is directed if for every X, Y ∈ Obj(C) there exist W ∈ Obj(C) and C-arrows f : W → X, g : W → Y.

Definition

We say that C has the almost amalgamation property if for every ε > 0, for every C-arrows f : X → Z, g : Y → Z, there exist C-arrows f ′ : W → X and g′ : W → Y such that the diagram Y W Z X

g g′ f ′ f

is ε-commutative.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 13 / 24

Existence

Theorem

Assume K ⊆ CM+ and K-arrows are surjections.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 14 / 24

Existence

Theorem

Assume K ⊆ CM+ and K-arrows are surjections. Suppose K contains a dominating directed subcategory with the almost amalgamation property and with countably many objects.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 14 / 24

Existence

Theorem

Assume K ⊆ CM+ and K-arrows are surjections. Suppose K contains a dominating directed subcategory with the almost amalgamation property and with countably many objects. Then there exists a K-generic object.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 14 / 24

Some Fraïssé theory

Definition

Let K be as above. We say that K is a compact Fraïssé category if

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 15 / 24

Some Fraïssé theory

Definition

Let K be as above. We say that K is a compact Fraïssé category if K is directed,

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 15 / 24

Some Fraïssé theory

Definition

Let K be as above. We say that K is a compact Fraïssé category if K is directed, K has the almost amalgamation property,

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 15 / 24

Some Fraïssé theory

Definition

Let K be as above. We say that K is a compact Fraïssé category if K is directed, K has the almost amalgamation property, K has countably many objects, up to homeomorphisms.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 15 / 24

Some Fraïssé theory

Definition

Let K be as above. We say that K is a compact Fraïssé category if K is directed, K has the almost amalgamation property, K has countably many objects, up to homeomorphisms.

Definition

Let u = Ui, uj

i , ω in K is called Fraïssé if

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 15 / 24

Some Fraïssé theory

Definition

Let K be as above. We say that K is a compact Fraïssé category if K is directed, K has the almost amalgamation property, K has countably many objects, up to homeomorphisms.

Definition

Let u = Ui, uj

i , ω in K is called Fraïssé if

1

For every X ∈ Obj(K) there are n ∈ ω and a K-arrow f : Un → X.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 15 / 24

Some Fraïssé theory

Definition

Let K be as above. We say that K is a compact Fraïssé category if K is directed, K has the almost amalgamation property, K has countably many objects, up to homeomorphisms.

Definition

Let u = Ui, uj

i , ω in K is called Fraïssé if

1

For every X ∈ Obj(K) there are n ∈ ω and a K-arrow f : Un → X.

2

For every ε > 0, for every n ∈ ω, for every K-arrow f : Y → Un, there exist m > n and a K-arrow g : Um → Y that is ε-close to the bonding arrow um

n : Um → Un.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 15 / 24

Main results

Theorem

Let K be a compact Fraïssé category. Then there exists a Fraïssé sequence in K.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 16 / 24

Main results

Theorem

Let K be a compact Fraïssé category. Then there exists a Fraïssé sequence in K.

Theorem

Let u be a Fraïssé sequence in K, U = lim ← −

• u. Then U is K-generic.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 16 / 24

Back to the pseudo-arc

Lemma

I, the category of all continuous surjections of the unit interval, is a compact Fraïssé category.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 17 / 24

Back to the pseudo-arc

Lemma

I, the category of all continuous surjections of the unit interval, is a compact Fraïssé category.

Proof.

Use the Mountain Climbing Theorem.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 17 / 24

Back to the pseudo-arc

Lemma

I, the category of all continuous surjections of the unit interval, is a compact Fraïssé category.

Proof.

Use the Mountain Climbing Theorem.

Theorem

Let u be a Fraïssé sequence in I. Then lim ← − u is the pseudo-arc.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 17 / 24

Back to the pseudo-arc

Lemma

I, the category of all continuous surjections of the unit interval, is a compact Fraïssé category.

Proof.

Use the Mountain Climbing Theorem.

Theorem

Let u be a Fraïssé sequence in I. Then lim ← − u is the pseudo-arc.

Corollary (Irwin & Solecki 2006)

Let P be a chainable continuum. Then P is homeomorphic to the pseudo-arc if and only if for every ε > 0 for every continuous surjections f : P → I, g : I → I there exists a continuous surjection h: P → I such that g ◦ h is ε-close to f.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 17 / 24

Categories of retractions

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 18 / 24

Categories of retractions

Proposition

Let K be the category whose objects are all nonempty compact metric spaces and arrows are all right-invertible continuous mappings. Then there is no K-generic space.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 18 / 24

Categories of retractions

Proposition

Let K be the category whose objects are all nonempty compact metric spaces and arrows are all right-invertible continuous mappings. Then there is no K-generic space.

Proof.

Use Cook’s continuum. For details, see

• A. Całka, Skracanie produktów topologicznych, MSc, Uniwersytet

Warszawski, 2008.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 18 / 24

Example

Fix n ∈ ω ∪ {∞}. Let Dn (Dc

n) be the category whose objects are

nonempty (connected) polyhedra of dimension n, and arrows are continuous retractions.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 19 / 24

Example

Fix n ∈ ω ∪ {∞}. Let Dn (Dc

n) be the category whose objects are

nonempty (connected) polyhedra of dimension n, and arrows are continuous retractions.

Lemma

Dn / Dc

n are compact Fraïssé categories.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 19 / 24

Example

Fix n ∈ ω ∪ {∞}. Let Dn (Dc

n) be the category whose objects are

nonempty (connected) polyhedra of dimension n, and arrows are continuous retractions.

Lemma

Dn / Dc

n are compact Fraïssé categories.

Corollary

There exist a Dn-generic and a Dc

n-generic space.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 19 / 24

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 20 / 24

Example

Let S be the category whose objects are finite-dimensional simplices and arrows are affine surjections.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 21 / 24

Example

Let S be the category whose objects are finite-dimensional simplices and arrows are affine surjections.

Theorem (A. Kwiatkowska & W.K.)

S is a compact Fraïssé category and its Fraïssé limit is the Poulsen simplex, the unique metrizable simplex whose set of extreme points is everywhere dense.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 21 / 24

Example

Let S be the category whose objects are finite-dimensional simplices and arrows are affine surjections.

Theorem (A. Kwiatkowska & W.K.)

S is a compact Fraïssé category and its Fraïssé limit is the Poulsen simplex, the unique metrizable simplex whose set of extreme points is everywhere dense.

Corollary

The Poulsen simplex is S-generic.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 21 / 24

Example

Let S be the category whose objects are finite-dimensional simplices and arrows are affine surjections.

Theorem (A. Kwiatkowska & W.K.)

S is a compact Fraïssé category and its Fraïssé limit is the Poulsen simplex, the unique metrizable simplex whose set of extreme points is everywhere dense.

Corollary

The Poulsen simplex is S-generic. Parallel results concerning the Lelek fan:

• A. Kwiatkowska, W. Kubi´

s, The Lelek fan and the Poulsen simplex as Fraïssé limits, preprint, 2015.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 21 / 24

Embeddings as Fraïssé limits

Example (W. Bielas, M. Walczy´ nska, W.K.)

Fix a compact 0-dimensional metric space A = ∅. Consider the following category FA.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 22 / 24

Embeddings as Fraïssé limits

Example (W. Bielas, M. Walczy´ nska, W.K.)

Fix a compact 0-dimensional metric space A = ∅. Consider the following category FA. The objects are continuous mappings f : A → s, where s is finite. An arrow from f : A → s to g : A → t is a continuous surjection p: t → s such that p ◦ g = f. t A s

p g f

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 22 / 24

Embeddings as Fraïssé limits

Example (W. Bielas, M. Walczy´ nska, W.K.)

Fix a compact 0-dimensional metric space A = ∅. Consider the following category FA. The objects are continuous mappings f : A → s, where s is finite. An arrow from f : A → s to g : A → t is a continuous surjection p: t → s such that p ◦ g = f. t A s

p g f

Lemma

FA is a Fraïssé category.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 22 / 24

Theorem (W. Bielas, M. Walczy´ nska, W.K.)

Let e: A → 2ω be a topological embedding such that e[A] is nowhere dense in 2ω. Then e is the Fraïssé limit of FA.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 23 / 24

Theorem (W. Bielas, M. Walczy´ nska, W.K.)

Let e: A → 2ω be a topological embedding such that e[A] is nowhere dense in 2ω. Then e is the Fraïssé limit of FA.

Corollary (Knaster & Reichbach)

Every homeomorphism between two closed nowhere dense subsets of 2ω extends to an auto-homeomorphism of 2ω.

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 23 / 24

References

• W. Kubi´

s, Metric-enriched categories and approximate Fraïssé limits, preprint, http://arxiv.org/abs/1210.6506

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 24 / 24